Two complementary views of the 1D signal — the plate eigenmode decomposition of a 2D reshape, and the nodal-point process formed by the signal's own zero-crossings.
The plate branch reshapes the signal into a 2D field and projects it onto the eigenmodes of a simply-supported rectangular Kirchhoff–Love plate (∇⁴w = λw with w = ∇²w = 0 on the boundary). The eigenmodes are φ_{mn} = sin(mπx/L)·sin(nπy/L) with eigenvalues λ_{mn} ∝ (m² + n²)² — literal Chladni-plate physics, where each φ_{mn}'s zero set is the nodal pattern along which sand collects. The biharmonic operator (the squared bracket) is what makes this a plate and not a membrane. The nodal branch is a random-matrix / multiscale analysis of the 1D signal's zero-crossings, treated as a vibrating-medium nodal process — Poisson predicts ⟨r⟩ ≈ 0.386 for uncorrelated gaps, GOE predicts ⟨r⟩ ≈ 0.536 for level-repelling spectra.
Mean consecutive spacing ratio r = min(sₙ, sₙ₊₁) / max(sₙ, sₙ₊₁) of zero-crossing intervals. Borrowed directly from random matrix theory: Poisson (uncorrelated crossings) gives ⟨r⟩ ≈ 0.386, GOE / Wigner (level-repelling, as in many-body quantum systems) gives ⟨r⟩ ≈ 0.536, and perfectly regular nodes give r = 1. Periodic waveforms saturate at 1.0 (Sine Wave, Square Wave, Van der Pol, every logistic periodic window), chaos sits near GOE (Logistic Edge-of-Chaos at 1.0 reflects near-regularity of the window), and noisy/clustered signals sit well below. The most discriminating metric in the geometry (F=8.61).
log(1 + Fano factor) of zero-crossing intervals; Fano = variance / mean, so Poisson gives F=1 and bursty crossings drive it upward. Devil's Staircase (8.96), fBm Persistent (8.41), ETH/BTC Ratio (8.18), and Perlin Noise (7.89) all have heavy zero-crossing burstiness — long quiet stretches punctuated by rapid flips. Correlates strongly with Möbius-S³:phi_return_cv (+0.817), flagging the same clustering signature from a different angle.
Power-law exponent of zero-crossing count versus bandpass center frequency across six octave bands. Measures how pattern complexity grows with the "driving frequency" — the 1D analogue of how Chladni figures become more intricate at higher resonant modes. Weak discriminator (F=1.03, rank low) but picks out sources where the bandpassed structure genuinely scales across octaves versus signals whose spectra are concentrated in a narrow band.
Kolmogorov-Smirnov distance of positive/negative domain lengths from an exponential distribution. For Poisson crossings, domain lengths are exponential; structured signals deviate. Devil's Staircase (0.89), Temperature Drift (0.79), Rudin-Shapiro (0.73), LIGO Hanford/Livingston (0.72/0.67) all have strongly non-exponential domain distributions — clumped or bimodal — consistent with regime-switching or structured modulation. Discovered by ShinkaEvolve (chladni v1, direction #6).
Fraction of plate modal energy in the low-eigenvalue (λ ≤ median) half of the spectrum. Smooth fields sit in low-order modes (high score); high-frequency or scale-localized fields push energy into the upper half. Takagi Function (0.996), fBm Persistent (0.996), Riemann-Hardy-Littlewood (0.996), Minkowski ?(x) (0.994), and ETH/BTC Ratio (0.994) score highest — all signals whose reshaped 2D fields are dominated by smooth low-order plate modes. Logistic Period-2, Period-3, Period-5, and constants collapse to 0. The literal biharmonic-plate decomposition; sister metric plate_modal_entropy is class-only because it duplicates Spectral Analysis:spectral_entropy at r=0.93 (the plate eigenbasis IS the 2D DST).
Gini coefficient of the octave-band envelope binned over time. Measures time-frequency localization / intermittency: high when bandpass energy is concentrated in a few time bins, near zero when it's spread evenly. Forest Fire (0.65), Speech "Five" (0.57), Rössler Hyperchaos (0.53), Lotka-Volterra (0.53), and Quantum Walk (0.52) score highest. Temperature Drift, Respiration Waveform, both LIGO channels, and Pressure all sit at 0 (stationary bandpass envelopes). Surfaced by PCA-driven embedding discovery and retained for empirical discrimination of intermittent vs. stationary signals.
Fraction of signal variance falling inside the six octave bandpass bands. Distinguishes signals whose energy lives in the octave grid (most physical signals → 1.0) from signals whose energy is concentrated outside it. The four Torus Quasiperiodic sources and ARMA(2,1) all hit 1.0. Primes, Partition Function, Minkowski ?(x), and constants sit at 0 — these monotone or trivial sources put no power in the bandpass grid. Orthogonal to temporal_burstiness (which asks where in time the captured power lives) and to the plate metrics (which work on the 2D reshape).
| Source | Origin | Value |
|---|---|---|
| ARMA(2,1) | stochastic-process | 1.0000 |
| Bearing Normal | machine-vibration | 1.0000 |
| LFSR (16-bit) | algorithmic-bytes | 1.0000 |
| ··· | ||
| Sine Wave | signal-synthesis | 0.0001 |
| Takagi Function | special-functions | 0.0006 |
| Magnetic Pendulum (3-Magnet) | continuous-flows | 0.0006 |
| Source | Origin | Value |
|---|---|---|
| Temperature Drift | atmospheric | 0.7925 |
| Respiration Waveform | physiological | 0.7624 |
| Rudin-Shapiro | symbolic-dynamics | 0.7272 |
| ··· | ||
| OTOC Growth | random-matrix-quantum | 0.0000 |
| Random Telegraph | stochastic-process | 0.0432 |
| Random Steps | stochastic-process | 0.0559 |
| Source | Origin | Value |
|---|---|---|
| Quantum Walk | random-matrix-quantum | 2.1584 |
| Sine Map (Feigenbaum) | iterated-maps | 2.1481 |
| Periodic Word | symbolic-dynamics | 2.0618 |
| ··· | ||
| Damped Pendulum | continuous-flows | -0.6527 |
| Duffing Oscillator | continuous-flows | -0.5824 |
| 2-Torus Quasiperiodic | coupled-oscillators | -0.4916 |
| Source | Origin | Value |
|---|---|---|
| Spectral Form Factor | random-matrix-quantum | 8.7156 |
| fBm (Persistent) | stochastic-process | 8.4143 |
| Devil's Staircase | special-functions | 8.2619 |
| ··· | ||
| Sine Map (Feigenbaum) | iterated-maps | 0.0000 |
| Logistic r=3.5 (Period-4) | iterated-maps | 0.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Sine Map (Feigenbaum) | iterated-maps | 1.0000 |
| Logistic r=3.2 (Period-2) | iterated-maps | 1.0000 |
| Logistic r=3.5 (Period-4) | iterated-maps | 1.0000 |
| ··· | ||
| OTOC Growth | random-matrix-quantum | 0.0000 |
| Mian-Chowla | number-theory | 0.0337 |
| Spectral Form Factor | random-matrix-quantum | 0.0719 |
| Source | Origin | Value |
|---|---|---|
| Takagi Function | special-functions | 0.9957 |
| fBm (Persistent) | stochastic-process | 0.9955 |
| Riemann-Hardy-Littlewood | number-theory | 0.9955 |
| ··· | ||
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0000 |
| Logistic r=3.83 (Period-3 Window) | iterated-maps | 0.0012 |
| Logistic r=3.74 (Period-5 Window) | iterated-maps | 0.0026 |
| Source | Origin | Value |
|---|---|---|
| Aubry-André Critical | random-matrix-quantum | 0.7671 |
| Devil's Staircase | special-functions | 0.7470 |
| Forest Fire | self-organized-criticality | 0.6616 |
| ··· | ||
| Critical Transition (Fold) | iterated-maps | 0.0000 |
| Sine Wave | signal-synthesis | 0.0000 |
| Brownian Walk | stochastic-process | 0.0000 |
Chladni asks two parallel questions: how does the signal decompose onto plate eigenmodes (the literal physics), and what are the statistics of its zero-crossings (the nodal process)? The plate branch (plate_low_mode_fraction, captured_power_fraction) sorts signals by smoothness and by whether their energy fits the octave grid. The nodal branch (nodal_gap_ratio, nodal_clustering, modal_nodal_cascade, domain_ks_exponential) sorts the atlas cleanly by dynamical class: periodic waveforms have regular spacings (nodal_gap_ratio → 1), chaotic attractors have GOE-like level repulsion, and heavy-tailed clustered processes (financial ratios, 1/f noise, seismic data) show elevated clustering with non-exponential domain lengths. temporal_burstiness adds a time-frequency intermittency channel that complements both branches. Not a chaos detector per se — a gap-statistics + plate-modal detector that happens to place chaotic systems in a different part of the plane than periodic or stochastic ones.